## Multiplying Polynomials #1 - Multiplying with Monomials

## Introduction

A monomial is an algebraic expression containing a single term. A polynomial is an algebraic expression containing multiple terms. Below are some examples of monomials and polynomials.

A polynomial with two terms is called a ** binomial** and one with three terms is a

**.**

*trinomial*

Most new topics in mathematics are introduced with the easiest problems first followed by the more complicated problems later. The problems in this lesson involve multiplying two expressions, one of them being a monomial. It is important that you completely understand this beginning material before moving on to more complicated polynomial multiplication.

## Lesson

This is the first of several lessons on the topic of multiplying polynomials. Since a polynomial is comprised of several individual terms, the ** distributive property** is used to do the multiplication.

In the problem to the left, the 6 is distributed to each term in the trinomial 2a^{2} + 5a + 3. Once the 6 is distributed to each term in the trinomial, the multiplication that follows is actually rather easy.

In the problem to the right, the term that is distributed is the 8b. The multiplication here is slightly more difficult because you must multiply the 8 times the number in each term and the b times the variable in each term.

This third problem is the first problem in this lesson that contains a minus sign in the polynomial. The problem can be worked in much the same way as the first two problems in this lesson. Notice that the minus sign remains in the same position throughout the work and in the answer.

When the negative sign is found in the monomial, it must also be distributed to each term of the polynomial. Two examples of this are given so that you can get a good idea of exactly how to simplify any problem where a negative monomial is multiplied by a polynomial.

Once you get used to distributing the monomial, you may be able to successfully complete many these problems in your head. It is fine to do some mental math, but attempting more difficult problems without showing work is a trap that leads many students into making careless mistakes. When you multiply larger polynomials that contain negative signs, more coefficients, and larger exponents it is important that you show all your work. It only takes a few seconds and 1-2 extra lines for each problem, so showing work is an investment that you should make as often as you can. It will pay you back in getting more right answers and feeling better about your answers altogether. It is recommended that you practice showing your work even on problems that you can successfully complete in your head. Remember that “perfect” practice leads to more right answers and better test grades, which leads to more confidence in yourself and enjoyment of your learning experience.

## Try It

1) 5(2m^{2} – 3m + 6)

2) 2n^{2}(-4n^{3} + 9n^{2} + 3n – 10)

3) 3p(-2p^{5} – p^{4} + - 2p^{3} + 11p^{2} – 3p)

4) -5(2q^{3}+ 3q^{2} – 11q + 5)

5) -8r^{2}(14 + 4r^{2} – 3r^{3} – 5r)

(Scroll down for answers)

Answers:

1) 10m^{2} – 15m + 30

2) -8n^{5} + 18n^{4} + 6n^{3} – 20n^{2}

3) -6p^{6} – 3p^{5} + - 6p^{4} + 33p^{3} – 9p^{2}

4) -10q^{3 }– 15q^{2} + 55q – 25

5) -112r^{2} – 32r^{4} + 24r^{5} + 40r^{3} , which equals 24r^{5} – 32r^{4} + 40r^{3} – 112r^{2} in standard form.

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